Liebig's law of the minimum

Liebig's Law of the Minimum, often simply called Liebig's Law or the Law of the Minimum, is a principle developed in agricultural science by Carl Sprengel (1828) and later popularized by Justus von Liebig. It states that growth is controlled not by the total of resources available, but by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth, where it was found that increasing the amount of plentiful nutrients did not increase plant growth. Only by increasing the amount of the limiting nutrient (the one most scarce in relation to "need") was the growth of a plant or crop improved.

Liebig used the image of a barrel—now called Liebig's barrel—to explain his law. Just as the capacity of a barrel with staves of unequal length is limited by the shortest stave, so a plant's growth is limited by the nutrient in shortest supply.

Liebig's Law has been extended to biological populations (and is commonly used in ecosystem models). For example, the growth of an organism such as a plant may be dependent on a number of different factors, such as sunlight or mineral nutrients (e.g. nitrate or phosphate). The availability of these may vary, such that at any given time one is more limiting than the others. Liebig's Law states that growth only occurs at the rate permitted by the most limiting. For instance, in the equation below, the growth of population $O$ is a function of the minimum of three Michaelis-Menten terms representing limitation by factors $I$ , $N$ and $P$ .

$\frac{dO}{dt} = min \left( \frac{I}{k_{I} + I}, \frac{N}{k_{N} + N}, \frac{P}{k_{P} + P} \right)$

It is limited to a situation where there are steady state conditions, and factor interactions are tightly controlled.